Question

If x power 2+1/x power 2=98, then find value of (x power 3+1/x power 3)



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Answer 1

Question :

If $\bf{x^{2} + \dfrac{1}{x^{2}} = 98}$ , then find the value of $\bf{x^{3} + \dfrac{1}{x^{3}}}$

Solution :

By solving the equation , we get :

$:\implies \bf{x^{2} + \dfrac{1}{x^{2}} = 98}$

We know that ,

$\bf{a^{2} + b^{2} = (a + b)^{2} - 2ab}$

By using the above formula, we get :

$:\implies \bf{\bigg(x + \dfrac{1}{x}\bigg)^{2} - 2 \times x \times \dfrac{1}{x} = 98}$

$:\implies \bf{\bigg(x + \dfrac{1}{x}\bigg)^{2} - 2 \times \not{x} \times \dfrac{1}{\not{x}} = 98}$

$:\implies \bf{\bigg(x + \dfrac{1}{x}\bigg)^{2} - 2 = 98}$

$:\implies \bf{\bigg(x + \dfrac{1}{x}\bigg)^{2} = 98 + 2}$

$:\implies \bf{\bigg(x + \dfrac{1}{x}\bigg)^{2} = 100}$

$:\implies \bf{x + \dfrac{1}{x} = \sqrt{100}}$

$:\implies \bf{x + \dfrac{1}{x} = 10}$

$\boxed{\therefore \bf{x + \dfrac{1}{x} = 10}}$⠀⠀⠀⠀⠀⠀⠀Eq.(i)

Now by cubing the equation (i) , we get :

$:\implies \bf{\bigg(x + \dfrac{1}{x}\bigg)^{3} = 10^{3}}$

We know that ,

$\bf{(a + b)^{3} = a^{3} + b^{3} + 3ab(a + b)}$

By using the above formula, we get :

$:\implies \bf{x^{3} + \bigg(\dfrac{1}{x}\bigg)^{3} + 3 \times x \times \dfrac{1}{x}\bigg(x + \dfrac{1}{x}\bigg) = 10^{3}}$

$:\implies \bf{x^{3} + \bigg(\dfrac{1}{x}\bigg)^{3} + 3 \times \not{x} \times \dfrac{1}{\not{x}}\bigg(x + \dfrac{1}{x}\bigg) = 10^{3}}$

$:\implies \bf{x^{3} + \bigg(\dfrac{1}{x}\bigg)^{3} + 3\bigg(x + \dfrac{1}{x}\bigg) = 10^{3}}$

We know the value of x + 1/x (i.e, 10) , so by putting it in the Equation , we get :

$:\implies \bf{x^{3} + \bigg(\dfrac{1}{x}\bigg)^{3} + 3 \times 10 = 10^{3}}$

$:\implies \bf{x^{3} + \bigg(\dfrac{1}{x}\bigg)^{3} + 30 = 10^{3}}$

$:\implies \bf{x^{3} + \bigg(\dfrac{1}{x}\bigg)^{3} + 30 = 1000}$

$:\implies \bf{x^{3} + \dfrac{1}{x^{3}} = 1000 - 30}$

$:\implies \bf{x^{3} + \dfrac{1}{x^{3}} = 970}$

$\boxed{\therefore \bf{x^{3} + \dfrac{1}{x^{3}} = 970}}$

Hence the value of $\bf{x^{3} + \dfrac{1}{x^{3}}}$ is 970.

Answer 2

Solution:

• x² + 1/x² = 98 ... given ... (1)

Note:

• x (1/x) = 1

» x² + 1/x² = 98

Add 2 on both sides

» x² + 2(x)(1/x) + 1/x² = 100

» (x + 1/x)² = 100

» x + 1/x = 10 ... (2)

Now, x³ + 1/x³ = (x + 1/x) (x² - (x) (1/x) + 1/x²)

from (1) and (2),

» x³ + 1/x³ = 10 (98 - 1)

» x³ + 1/x³ = 970